arXiv subjects

Jevgenijs Ivanovs

Publications and source records attributed to Jevgenijs Ivanovs.

At least 19 recordsLinked to original sources

arXiv2024

Lévy graphical models

Conditional independence and graphical models are crucial concepts for sparsity and statistical modeling in higher dimensions. For Lévy processes, a widely applied class of stochastic processes, these notions have not been studied. By the Lévy-Itô decomposition, a multivariate Lévy process can be decomposed into the sum of a Brownian motion part and an independent jump process. We show that conditional independence statements between the marginal processes can be studied separately for these two parts. While the Brownian part is well-understood, we derive a novel characterization of conditional independence between the sample paths of the jump process in terms of the Lévy measure. We define Lévy graphical models as Lévy processes that satisfy undirected or directed Markov properties. We prove that the graph structure is invariant under changes of the univariate marginal processes. Lévy graphical models allow the construction of flexible, sparse dependence models for Lévy processes in large dimensions, which are interpretable thanks to the underlying graph. For trees, we develop statistical methodology to learn the underlying structure from low- or high-frequency observations of the Lévy process and show consistent graph recovery. We apply our method to model stock returns from U.S. companies to illustrate the advantages of our approach.

math.ST↗

arXiv2024

Graphical models for infinite measures with applications to extremes

Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure $Λ$ on the punctured Euclidean space $\mathbb R^d\setminus \{0\}$ that explodes at the origin. The importance of such measures stems from their connection to infinitely divisible and max-infinitely divisible distributions, where they appear as Lévy measures and exponent measures, respectively. We characterize independence and conditional independence for $Λ$ in various ways through kernels and factorization of a modified density, including a Hammersley-Clifford type theorem for undirected graphical models. As opposed to the classical conditional independence, our notion is intimately connected to the support of the measure $Λ$. Our general theory unifies and extends recent approaches to graphical modeling in the fields of extreme value analysis and Lévy processes. Our results for the corresponding undirected and directed graphical models lay the foundation for new statistical methodology in these areas.

math.ST↗

arXiv2024

One-sided Markov additive processes with lattice and non-lattice increments

Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a level variable is governed by an auxiliary phase variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is one-sided, that is it is jump-free in one direction. From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be lattice random variables and for fluid queues, they have to be piecewise linear. In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.

math.PR↗

arXiv2024

Distributionally robust halfspace depth

Tukey's halfspace depth can be seen as a stochastic program and as such it is not guarded against optimizer's curse, so that a limited training sample may easily result in a poor out-of-sample performance. We propose a generalized halfspace depth concept relying on the recent advances in distributionally robust optimization, where every halfspace is examined using the respective worst-case distribution in the Wasserstein ball of radius $δ\geq 0$ centered at the empirical law. This new depth can be seen as a smoothed and regularized classical halfspace depth which is retrieved as $δ\downarrow 0$. It inherits most of the main properties of the latter and, additionally, enjoys various new attractive features such as continuity and strict positivity beyond the convex hull of the support. We provide numerical illustrations of the new depth and its advantages, and develop some fundamental theory. In particular, we study the upper level sets and the median region including their breakdown properties.

math.OC↗

arXiv2022

A series expansion formula of the scale matrix with applications in change-point detection

We introduce a new Levy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.

stat.ME↗

arXiv2022

Probability of total domination for transient reflecting processes in a quadrant

We consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that such behaviour is certain and each component can dominate the other with positive probability for any given starting position. Additionally, we establish the corresponding invariance principle providing justification for the use of the reflected Brownian motion as an approximate model. Focusing on the probability that the first component dominates, we derive a kernel equation for the respective Laplace transform in the starting position. This is done for the compound Poisson model with negative exponential jumps and, by means of approximation, for the Brownian model. Both equations are solved via boundary value problem analysis, which also yields the domination probability when starting at the origin. Finally, certain asymptotic analysis and numerical results are presented.

math.PR↗

arXiv2021

Implementable coupling of Lévy process and Brownian motion

We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of increments. Applications to model risk and multilevel Monte Carlo are discussed in detail, and numerical examples are provided.

math.PR↗

arXiv2021

Lévy processes conditioned to stay in a half-space with applications to directional extremes

This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

math.PR↗

arXiv2021

On scale functions for Lévy processes with negative phase-type jumps

We provide a novel expression of the scale function for a Lévy processes with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows to employ phase-type distributions with a hundred of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

math.PR↗

arXiv2021

Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.

math.PR↗

arXiv2020

Recovering Brownian and jump parts from high-frequency observations of a Lévy process

We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a carefully chosen threshold, in which case both methods yield the same rate of convergence. This rate depends on the small-jump activity and is given in terms of the Blumenthal-Getoor index. Finally, we discuss possible extensions, including the multidimensional case, and provide numerical illustrations.

math.PR↗

arXiv2020

Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid

For a Lévy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/α$-self-similar Lévy process with $α\in(0,2]$, and restrict to $α>1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.

math.PR↗

arXiv2020

Discretization of the Lamperti representation of a positive self-similar Markov process

This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

math.PR↗

arXiv2020

Sparse Structures for Multivariate Extremes

Extreme value statistics provides accurate estimates for the small occurrence probabilities of rare events. While theory and statistical tools for univariate extremes are well-developed, methods for high-dimensional and complex data sets are still scarce. Appropriate notions of sparsity and connections to other fields such as machine learning, graphical models and high-dimensional statistics have only recently been established. This article reviews the new domain of research concerned with the detection and modeling of sparse patterns in rare events. We first describe the different forms of extremal dependence that can arise between the largest observations of a multivariate random vector. We then discuss the current research topics including clustering, principal component analysis and graphical modeling for extremes. Identification of groups of variables which can be concomitantly extreme is also addressed. The methods are illustrated with an application to flood risk assessment.

stat.ME↗

arXiv2020

Optimal estimation of some random quantities of a Lévy process

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time measure of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators. Furthermore, we discuss pre-estimation of the parameters of the underlying models, which is required for practical implementation of the proposed statistics.

math.PR↗

arXiv2019

Stochastic decompositions in bivariate risk and queueing models with mutual assistance

We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochastic decomposition. Each one can be seen as an independent sum of respective quantities for the two models with one-way interaction. Additionally, we discuss a rather general method of establishing decompositions from a given kernel equation by identifying two independent random variables from their difference, which may be useful for other models. Finally, we point out that the same decomposition is true for uncorrelated Brownian motion reflected to stay in an orthant, and it concerns the face measures appearing in the basic adjoint relationship.

math.PR↗

arXiv2018

A factorization of a Lévy process over a phase-type horizon

This note provides a factorization of a Lévy pocess over a phase-type horizon $τ$ given the phase at the supremum, thereby extending the Wiener-Hopf factorization for $τ$ exponential. One of the factors is defined using time reversal of the phase process. It is shown that there are a variety of time-reversed representations, all yielding the same factor. Consequences of this are discussed and examples provided. Additionally, some explicit formulas for the joint law of the supremum and the terminal value of the process at $τ$ are given.

math.PR↗

arXiv2018

Discretization error for a two-sided reflected Lévy process

An obvious way to simulate a Lévy process $X$ is to sample its increments over time $1/n$, thus constructing an approximating random walk $X^{(n)}$. This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resultant process $Y$ and regulators $L,U$ at the lower and upper barriers at some fixed time. Under the weak assumption that $X_\varepsilon/a_\varepsilon$ has a non-trivial weak limit for some scaling function $a_\varepsilon$ as $\varepsilon\downarrow 0$, it is proved in particular that $(Y_1-Y^{(n)}_n)/a_{1/n}$ converges weakly to $\pm V$, where the sign depends on the last barrier visited. Here the limit $V$ is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (2017). Some further insight in the distribution of $V$ is provided both theoretically and numerically.

math.PR↗